Deligne–Drinfeld conjecture for grt₁

Prove that the injection from the free Lie algebra generated by the symbols σ₃, σ₅, σ₇, … into the Grothendieck–Teichmüller Lie algebra grt₁ is an isomorphism; i.e., establish that grt₁ is freely generated by the classes σ_{2k+1} with loop order 2k+1.

Background

The paper recalls that H⁰(GC₂) is identified with the Grothendieck–Teichmüller Lie algebra grt₁, and that Brown constructed an injective Lie algebra map from a free Lie algebra generated by symbols σ₃, σ₅, σ₇, … (with σ_{2k+1} of loop order 2k+1).

The Deligne–Drinfeld conjecture asserts that this injection is an isomorphism. The authors note numerical verification up to loop order 29, but no general proof is known.